PreprintCC0arXiv
Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients
Abstract
In this paper we obtain sharp exponents in the Aleksandrov-Bakelman-Pucci maximum estimate for linear uniformly elliptic equations with measurable coefficients, originally conjectured by Pucci in 1966 \cite{Pucci}, together with corresponding sharp results under the more natural quantification of uniform ellipticity in more than two dimensions. The proofs combine a Hessian integral inequality obtained from boundary Morse inequalities with an obstacle argument. The estimates also hold on arbitrary open sets, of finite volume, without regularity assumptions on their boundaries.
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BibTeX
@misc{chen2026sharp,
title = {{Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients}},
author = {Shibing Chen and Neil S. Trudinger and Xu-Jia Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23596},
eprint = {2610.23596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23596},
note = {CC0},
}
RIS
TY - UNPB
TI - Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients
AU - Shibing Chen
AU - Neil S. Trudinger
AU - Xu-Jia Wang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23596
UR - https://arxiv.org/abs/2610.23596
AB - In this paper we obtain sharp exponents in the Aleksandrov-Bakelman-Pucci maximum estimate for linear uniformly elliptic equations with measurable coefficients, originally conjectured by Pucci in 1966 \cite{Pucci}, together with corresponding sharp results under the more natural quantification of uniform ellipticity in more than two dimensions. The proofs combine a Hessian integral inequality obtained from boundary Morse inequalities with an obstacle argument. The estimates also hold on arbitrary open sets, of finite volume, without regularity assumptions on their boundaries.
ER -